Question

There are 10 cars in the lottery, each at a cost of $100,000. There are also 1,000 consolation prizes, $1000 each. The total number of the tickets is 10,000,000. The price of a ticket is $2. Find the mathematical expectation of the lottery.

Answer 1

• - $1.8

Step-by-step explanation:

The mathematical expectation is the sum of the products of the of each outcome by its probability.

Calling E(X) the mathematical expectation, that is:

`E(X)=\sum\limits^n_1 {x_i\cdot P(x_i)}`

The possible outcomes and its probability are:

1) For the cars

• Probability: number of cars / number of tickets

• Probability = 10/10,000,000 = 1/1,000,000

• Value of each car = $100,000

2) For the consolation prizes

• Probability = number of consolation prizes / number of tickets

• Probability = 1,000 / 10,000,000 = 1/10,000

• Value of each consolation prize = $1,000

3) Mathematical expectation of the prizes

• E(X) = $100,000 × 1/1,000,000 + $1,000 × 1/10,000 = $0.2

3) Cost

Now, you must subtract the price of a ticket, since it is a certain cost, this is, it has 100% probability.

• Price of the ticket : $2

4) Mathematical expectation of the lottery

It is the expectation of the prizes less the cost:

• $0.2 - $2 = - $1.8 ← answer